A construction of a 𝜆-Poisson generic sequence

Becher, Verónica; Sac Himelfarb, Gabriel

doi:10.1090/mcom/3806

A construction of a $\lambda$-Poisson generic sequence
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by Verónica Becher and Gabriel Sac Himelfarb HTML | PDF

Math. Comp. 92 (2023), 1453-1466 Request permission

Abstract:

Years ago Zeev Rudnick defined the $\lambda$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $\lambda$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $\lambda$-Poisson generic sequence over any alphabet and any positive $\lambda$, except for the case of the two-symbol alphabet, in which it is required that $\lambda$ be less than or equal to the natural logarithm of $2$. Since $\lambda$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $\lambda$-Poisson generic.

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